From "The Poetical Geography" by George Van Waters, 1857
Square root
Divide into periods of two figures each,
The number you know, as the pedagogues teach, -
In the left hand period find the greatest square,
Which from it subtract, and to what remains there
Bring the next period down for a Dividend (fair):
Place the root of the square at the right hand of all,
And two times the root a Divisor we call.
Then try the Divisor, see how many times
The Dividend holds it (by prose or by rhymes).
Of its right hand figure exclusive, you know,
And write in the root the number't will go,
Then to the Divisor the same figure tie,
And by the same figure the whole multiply;
The product then take from the Dividend (penned),
And of that which remains, make a new dividend;
By bringing the period that's next, along side,-
And for a Divisor that's new and untried,
Just double the figures that stand in the root,
And work as before, till the answer is got.
For a more useful discussion of this procedure, read Extracting Square Roots with pencil and paper.
Another Poem
Here is a nice explanation of the different methods used in the past: Ancient Methods of Calculating Square Roots.
Included in this article is an image from a textbook written in 1772. I have reproduced that image here as text:
Square Roots "made plain and easy"
The root of your first period you
Must place in quote, if you work true;
Whose square from your said period then
You must subtract; and to the remain
Another period being brought,
You must divide as here is taught,
By the double of your quote, but see
Your unit's place you do leave free;
Which place will be supplied by the Square
Of your next quoted figure there:
Next multiply, subtract, and then
Repeat your work unto the end;
And if your number be irrational,
Add pairs of cyphers for a decimal.